Discretely ordered groups

TL;DR

This paper investigates conditions under which orderable groups can have discrete orders, showing embeddings and providing examples including braid groups and non-locally-indicable groups.

Contribution

It establishes necessary and sufficient conditions for groups to admit discrete orders and demonstrates embeddings of all orderable groups into discretely orderable groups.

Findings

Free groups cannot have discrete orders.

Every orderable group embeds into a discretely orderable group.

Examples include Artin braid groups and Bergman's non-locally-indicable groups.

Abstract

We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group G embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman's non-locally-indicable right orderable groups.